Distance between orbits

Forum,
I'm addressing the problem of computing the minimum possible distance between two non-interacting bodies on elliptical orbits. From a general point of view, it looks like a minimization problem of a function of two variables, e.g. in the domain [0,2*pi)*[0,2*pi). This problem can be numerically addressed in a standard fashion, e.g. by a conjugate gradient method. But I wonder if an analytical approach exists that can simplify the problem - maybe reducing it to unidimensional - and significantly speed-up the computation.

Forum,
I'm addressing the problem of computing the minimum possible distance between two non-interacting bodies on elliptical orbits. From a general point of view, it looks like a minimization problem of a function of two variables, e.g. in the domain [0,2*pi)*[0,2*pi). This problem can be numerically addressed in a standard fashion, e.g. by a conjugate gradient method. But I wonder if an analytical approach exists that can simplify the problem - maybe reducing it to unidimensional - and significantly speed-up the computation.

For this problem, you don't need to know where the planet is in its orbit at any particular time, so you can ignore the epoch of mean anomaly and time of perihelion passage. But you do still need the other elements ( a, e, i, L, w ).

You probably know this method already, judging by what you wrote. I used the plain old peck-peck-peck method, except I added an outer loop for homing in on the part of the barnyard where the feed is the thickest. It might speed things up a little.

In the procedure below, for variables having two subscripts, the first subscript will designate which orbit (either 0 or 1) and the second subscript will designate either the beginning (0) or the end (1) of an eccentric anomaly search interval. For variables having only one subscript, the subscript will specify which orbit. All eccentric anomalies, and all other angles, are used only in radians.